Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_194

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Regard the tall building of Prob. 5/193 as a uniform upright beam. Determine and plot the shear force and bending moment in the structure as functions of the height above the ground. Evaluate your expressions at . Context from Problem 5/193: A 900-ft building is subjected to a wind load with a parabolic pressure distribution . At the top (), the pressure is . The building depth perpendicular to the paper is . Problem 5/194 (a) Functional expressions for shear force and bending moment 1. Formula: First, we determine the pressure constant and the distributed load intensity . Then, the shear force and bending moment at height are found by integrating the loads from to the top of the building (). 2. Substitution: From Prob. 5/193: at , . With depth : Substitute into integrals: x x=450 ft p=k xx=900 ft 12 lb/ft 2 200 ft k w(x)V(x)M(x)x xH=900 ft p(x)=k x w(x)=p(x)⋅b V(x)= w ( ξ )d ξ∫ x H M(x)= w ( ξ )( ξ − ∫ x H x)dξ x=900 ftp= 12 lb/ft 2 12=k ⟹ 900k= = 30 12 0.4 lb/ft 2.5 b=200 ft w(x)=(0.4 )⋅x200= 80 x lb/ft 1/2 3. Calculation: For shear force: For bending moment: 4. Result: ● Final Conclusion: The shear force and bending moment are functions of height (in feet) given by and . (b) Evaluation at 1. Formula: Substitute into the derived functional expressions for and . V(x)= 80 ξ d ξ∫ x 900 1/2 M(x)= 80 ξ ( ξ − ∫ x 900 1/2 x)dξ V(x)= 80 = [ 1.5 ξ 3/2 ] x 900 (900− 3 160 3/2 x )= 3/2 (27,000− 3 160 x ) 3/2 V(x)=1,440,000− x lb 3 160 3/2